1. Mathematics, Proofs and Computation
Madhu Sudan
Harvard
January 4, 2016
IIT-Bombay: Math, Proofs, Computing

2. Logic, Mathematics, Proofs
Reasoning:
Start with body of knowledge.
Add to body of knowledge by new observations, and new deductions
Process susceptible to errors:
One erroneous observation may propagate.
Constant process of “consistency checking”.
Mathematics = Language of Precision
Captures (subset of) knowledge precisely.
Proofs: Enable checking of consistency of precisely stated facts.
January 4, 2016
IIT-Bombay: Math, Proofs, Computing

3. In this talk: Proofs and Computation
“Computer Assisted Proofs ?”
[Appel-Haken] – 4-color theorem
[Hales] – Kepler Conjecture
[Petkovsky,Wilf,Zeilberger] – “A=B”
No!
Mathematics
Computing
Proofs
January 4, 2016
IIT-Bombay: Math, Proofs, Computing

4. IIT-Bombay: Math, Proofs, Computing
Formal Logic
Attempts to convert reasoning to symbolic manipulation.
Remarkably powerful.
Originated independently, and with different levels of impact, in different civilizations …
Mathematics
Proofs
"Aristotle Altemps Inv8575" by Copy of Lysippus - Jastrow (2006). Licensed under Public Domain via Commons -
January 4, 2016
IIT-Bombay: Math, Proofs, Computing

5. IIT-Bombay: Math, Proofs, Computing
George Boole ( )
Mathematics
Proofs
The strange math of ( 0,1 ; ∨,∧, ¬)
Typical Derivation:
Axiom: Repitition does not add knowledge
Formally: 𝑥𝑥=𝑥
Example: Object is Good and Good ≡ Object is Good
Consequence: Principle of Contradiction
“… it is impossible for any being to possess a quality and at the same time to not possess it.”
Proof: 𝑥 2 =𝑥⇒ 𝑥 2 −𝑥=0⇒𝑥 𝑥−1 =0
⇒𝑥=0 or ¬𝑥≝1−𝑥=0
⇒𝑥 or ¬𝑥 does not hold
(page 34)
January 4, 2016
IIT-Bombay: Math, Proofs, Computing

6. IIT-Bombay: Math, Proofs, Computing
ℝ, ℚ,ℤ vs. {𝟎,𝟏}
Boole’s Mathematics:
Focus on tiny part of mathematical universe.
Progress
In Math
ℝ: Algebra/Calculus
ℚ: optimization
ℤ: number theory
{0,1}
January 4, 2016
IIT-Bombay: Math, Proofs, Computing

7. Boole’s “modest” ambition
“The design of the following treatise is to investigate the fundamental laws of those operations of the mind by which reasoning is performed; to give expression to them in the symbolical language of a Calculus, and upon this foundation to establish the science of Logic and construct its method; to make that method itself the basis of a general method for the application of the mathematical doctrine of Probabilities; and, finally, to collect from the various elements of truth brought to view in the course of these inquiries some probable intimations concerning the nature and constitution of the human mind.” [G.Boole, “On the laws of thought …” p.1]
Mathematics ≼ All of reasoning ≼{0,1}
ℝ: Algebra/Calculus
ℚ: optimization
ℤ: number theory
{0,1}
January 4, 2016
IIT-Bombay: Math, Proofs, Computing

8. IIT-Bombay: Math, Proofs, Computing
Whither Computing?
How well does the logic capture mathematics?
Cantor‘1890: Logic may face some problems?
Hilbert ‘1900:
Should capture everything!
Godel ‘1920s:
Incompleteness
Church-Turing 1930s: Incompleteness holds for any effective reasoning procedure.
This statement
is not
provable
true
January 4, 2016
IIT-Bombay: Math, Proofs, Computing

9. Turing’s Machine Model of computer → von Neumann architecture
Mathematics
Proofs
Turing’s Machine
Computing
Model of computer
Encodings of other machines
- Universal!
→ von Neumann architecture
Finite
State
Control
RAM
Universal
Machine
CPU
R/W
One machine to rule them all!
January 4, 2016
IIT-Bombay: Math, Proofs, Computing

10. Proofs: Story so far Proof: Has to be mechanically verifiable.
Theorem: Statement with a proof.
Incompleteness: There exist statements consistent with the system of logic that do not admit a proof.
Unaddressed: What difference does proof make?
Theorem: 𝑇
Proof: Π 1
⇒ Π 2
⇒ Π 3
⇒ Π 4 …
⇒ Π ℓ =𝑇
Theorem: 𝑇
Proof:
Has ℓ lines
Both mechanically verifiable!
#steps ~ℓ
~ 2 ℓ
January 4, 2016
IIT-Bombay: Math, Proofs, Computing

11. Origins of Modern Complexity
[Gödel 1956] in letter to von Neumann: “Is there a more “effective” procedure to find proof of length ℓ if one exists?” (in ℓ 2 steps? ℓ ℓ 2 ?)
[Cobham, Edmonds, Hartmanis, Stearns – 60s]:
Time Complexity is a (coarse) measure. 10 ℓ 2 =5 ℓ 2 ! But ℓ 2 > ℓ 1.9 .
𝑃≝ problems solvable in time ℓ 𝑐 for constant 𝑐
Edmonds Conjecture: Travelling Salesman Problem is not solvable in 𝑃
January 4, 2016
IIT-Bombay: Math, Proofs, Computing

12. Proofs, Complexity & Optimization!
[Cook ’70]
Complexity of
Theorem Proving
[Levin ’71]
Universal Search problems
Formalized Edmond’s Conjecture:
𝑁𝑃= Problems w. efficiently verifiable solutions
𝑁𝑃-complete = Hardest problem in NP
Theorem-Proving NP-Complete
SAT (simple format of proofs) NP-complete
Domino tiling NP-Complete
Godel’s question ≡ “Is 𝑁𝑃=𝑃?”
January 4, 2016
IIT-Bombay: Math, Proofs, Computing

13. Proofs, Complexity & Optimization - 2
[Karp ‘72] Reducibility among
combinatorial optimization problems
Showed central importance of 𝑁𝑃.
Nineteen problems 𝑁𝑃-Complete!
Cover optimization, logic, combinatorics, graph theory, chip design.
January 4, 2016
IIT-Bombay: Math, Proofs, Computing

14. Some NP-complete Problems
Map Coloring: Can you color a given map with 3-colors, s.t. bordering states have diff. colors?
January 4, 2016
IIT-Bombay: Math, Proofs, Computing

15. Some NP-Complete Problems
Travelling Salesman Problem: (TSP) – Find tour of minimum length visiting given set of cities.
Image due to [Applegate, Bixby, Chvatal, Cook].
Optimal TSP visiting ~13000 most populated cities in US.
January 4, 2016
IIT-Bombay: Math, Proofs, Computing

16. Some NP-Complete Problems
Biology: Fold DNA sequence so as to minimize energy.
Economics: Finding optimal portfolio of stocks subject to budget constraint.
Industrial Engineering: Schedule tasks subject to precedence constraints to minimize completion time. …
January 4, 2016
IIT-Bombay: Math, Proofs, Computing

17. Consequences to Proof Checking
NP-Complete problem ≡ Format for proofs.
3-coloring is NP-complete ⇒ exists function 𝑓
𝑓 𝑇,ℓ = Map with ℓ 𝑐 regions s.t.
𝑇 has proof of length ℓ ⇒ Map is 3-colorable
… no proofs of length ℓ ⇒ Map not 3-colorable
Format?
Rather than convention proof, can simply give coloring of map!
Advantage: Error is local (two improperly colored regions)
Verifier computes 𝑓(𝑇,ℓ) and
verifies coloring is good
January 4, 2016
IIT-Bombay: Math, Proofs, Computing

18. Is P=NP? Don’t know … If P=NP … If P≠NP …
Cryptography might well be impossible (current systems all broken simultaneously)
All optimization problems become “easy”
… You get whatever you wish … if you can verify satisfaction.
Mathematicians replaced by computers.
If P≠NP …
… Consistent with current thinking, so no radical changes.
Proof would be very educational.
Might provide sound cryptosystems.
Independent of Peano’s axioms, Choice …?
“Of all the Clay Problems, this might be the one to find the shortest solution, by an amateur mathematician.”
- Devlin, The Millenium Problems (Possibly thinking P=NP)
“If someone shows P=NP, then they prove any theorem they wish. So they would walk away not just with $1M, but $6M by solving all the Clay Problems!”
- Lance Fortnow, Complexity Blog
“P = NP?” is Mathematics-Complete !!
January 4, 2016
IIT-Bombay: Math, Proofs, Computing

19. Post-Modern Complexity
Emphasis on Randomness.
Randomness can potentially speed up algorithms.
Essential for
Equilibrium behavior
Coordination among multiple players
Cryptography
But it probably can’t help with Logic – right?
Actually – it does!!
January 4, 2016
IIT-Bombay: Math, Proofs, Computing

20. IIT-Bombay: Math, Proofs, Computing
Interactive Proofs
[Goldwasser, Micali, Rackoff], [Babai] ~1985
Verifier asks questions and Prover responds:
Space of questions exponentially large in the length!
Prover has to be ready for all!
Many striking examples:
Pepsi ≠ Coke! (“Graphs not isomorphic”)
Can prove “theorem has no short proof”.
“IP = PSPACE” [LFKN, Shamir]
“Zero Knowledge Protocols” – Foundations of Secure communication
January 4, 2016
IIT-Bombay: Math, Proofs, Computing

21. Probabilistically Checkable Proofs
Do proofs have to be read in entirety to verify?
January 4, 2016
IIT-Bombay: Math, Proofs, Computing

22. Probabilistically Checkable Proofs
Do proofs have to be read in entirety to verify?
Conventional formats for proofs – YES!
But we can change the format!
Format ≡ Verification Algorithm
Any verifier is ok, provided:
If 𝑇 has proof of length ℓ in standard system, then 𝑉 should accept some proof of length poly(ℓ)
If 𝑇 has no proofs, then 𝑉 should not accept any proof
PCP Theorem [Arora, Lund, Motwani, Safra, Sudan, Szegedy ‘92]:
A format exists where V reads only
constant number of bits of proof!
with probability ≥ 1 2 X
January 4, 2016
IIT-Bombay: Math, Proofs, Computing

23. IIT-Bombay: Math, Proofs, Computing
PCPs and Optimization
Classical connection: [Cook → Karp]:
Solving optimization problems ≡ finding proofs
New Connection: [Feige et al., Arora et al.]
Solving optimization problems approximately ≡ finding nearly valid proofs.
Existence of nearly valid proofs ≡ Existence of perfectly valid proofs (due to PCPs)!
Conclude: Solving (some/many) optimizations approximately is as hard as solving them exactly!
1992-today: PCP-induced revolution in understanding approximability!!
January 4, 2016
IIT-Bombay: Math, Proofs, Computing

24. Summary and Conclusions
Computing is a science:
Goes to the very heart of scientific inquiry.
What big implications follow from local steps?
Search for proofs captures essence of all search and optimization.
“Is P=NP?” Central mathematical question.
Still open.
But lots of progress …
“Khot’s UGC” (Unique Games Conjecture): Cutting edge of optimization.
Khot’s Unique Games Conjecture: Cutting edge of optimization. Sharp thresholds almost everywhere?
Communication of Knowledge:
Will we use PCPs to communicate math proofs?
Will we use PCPs anywhere?
Computing is a science!!
What global changes can/can not be effected by sequence of local changes?
January 4, 2016
IIT-Bombay: Math, Proofs, Computing

25. IIT-Bombay: Math, Proofs, Computing
Thank You!
January 4, 2016
IIT-Bombay: Math, Proofs, Computing